API Score Project
David (Yongbock) Kwon
Predicting Academic Performance Index (API) Score
from California Department of Eduaction
through Linear Regression
library(car) #cross validation## Loading required package: carDatalibrary(ggplot2) #ggplot
library(lattice)
library(MASS)
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
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## select## The following object is masked from 'package:car':
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## recode## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, unionlibrary(caret) #linear
library(leaps) #regsubsets
library(glmnet)## Loading required package: Matrix## Loading required package: foreach## Loaded glmnet 2.0-16library(olsrr)##
## Attaching package: 'olsrr'## The following object is masked from 'package:MASS':
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## cement## The following object is masked from 'package:datasets':
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## riverslibrary(e1071)
library(sandwich)
library(tidyr)##
## Attaching package: 'tidyr'## The following object is masked from 'package:Matrix':
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## expandlibrary(purrr) #keep##
## Attaching package: 'purrr'## The following objects are masked from 'package:foreach':
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## accumulate, when## The following object is masked from 'package:caret':
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## lift## The following object is masked from 'package:car':
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## somelibrary(ModelMetrics) #mse##
## Attaching package: 'ModelMetrics'## The following object is masked from 'package:glmnet':
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## auc## The following objects are masked from 'package:caret':
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## confusionMatrix, precision, recall, sensitivity, specificity## The following object is masked from 'package:base':
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## kappalibrary(corrplot)## corrplot 0.84 loadedset.seed(1234)
Base05<- read.csv("/Users/DavidKwon/Desktop/Yongbock/API Score Prj/API 2005 Base Data.csv")Basic summary of the dataset from the website, California Department of Education
“The 2005 API (Academic Performance Index) summarizes a school’s, an LEA’s (local educational agency, is a school district or county office of education), or the state’s performance on the spring 2005 Standardized Testing and Reporting (STAR) Program and 2005 California High School Exit Examination (CAHSEE).”
“The 2005 API Base summarizes a subgroup’s (e.g STAR Program scores, Ethnic/racial subgroup/ parents’ degrees/ English Learners …) performance on the spring 2005 STAR Program and 2005 CAHSEE. It serves as the baseline score, or starting point, of performance of that subgroup”
Therefore, we are going to remove those variables that are created by our dependent variable such as SIM_RANK, ST_RANK
I am also going to remove one of the variable, RTYPE or STYPE because they are overlapping each other.
Base05<-select(Base05,c(-"RTYPE", -"SIM_RANK",-"ST_RANK"))Finding NAs rows of the dependent variables and removes the rows that includes NAs
Base05<-Base05[-which(is.na(Base05$API05B)),] Creating function that finds NAs for each variables
findingNA<-function(x){
length(which(is.na(x)))/length(x)
}
for(i in 1:dim(Base05)[2]){
if(findingNA(Base05[,i])>0.1){
print(i)
}
}## [1] 3
## [1] 4
## [1] 5
## [1] 42
## [1] 46
## [1] 47
## [1] 48
## [1] 66I am going to remove the variables that have 10% of the elements is NAs
I also removed the “X” variable (school or district code), which is 1st column
newbase<-Base05[,c(-1, -3, -4, -5, -42, -46, -47, -48, -66)]
new.base<-newbase[complete.cases(newbase),]lets explore our dependent variable with ggplot
Histogram of our dependent variable
ggplot(data=new.base, aes(new.base$API05B)) +
geom_histogram(aes(y=..density..), binwidth=10) +
geom_density(aes(y=..density..), color="red")+
labs(title="Histogram for Academic Performance Index 2005 Base", x="API 2005 Base", y="Frequency") +
geom_vline(xintercept = mean(new.base$API05B), show.legend = TRUE, color="red") +
geom_vline(xintercept = median(new.base$API05B), show.legend = TRUE, color="blue")
As the graph shows, we can notice the API05B seems to be left-skewed.
Our independent factor variables
new.base[1:97] %>% keep(is.factor) %>% gather() %>%
ggplot(aes(value))+
facet_wrap(~key,scales="free")+
geom_bar()## Warning: attributes are not identical across measure variables;
## they will be dropped
Our independent numerical variable
new.base[1:20] %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
new.base[21:40] %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
new.base[41:60] %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
new.base[61:80] %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
new.base[81:96] %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
Most of our independent numerical variables are highly skewed
We might want to perform log tranformation, but I would like to see the model before transformation, and if transformation is needed, I will perform it then.
Splitting training and test dataset - cross validation
split<-createDataPartition(y=new.base$API05B,p=0.7,list=FALSE)
training.newbase <- new.base[split,]
test.newbase <- new.base[-split,]The first model with whole data
lm<-lm(API05B~., data=training.newbase)
summary(lm)##
## Call:
## lm(formula = API05B ~ ., data = training.newbase)
##
## Residuals:
## Min 1Q Median 3Q Max
## -369.99 -23.30 0.19 23.70 387.98
##
## Coefficients: (15 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.060e+02 7.808e+02 0.648 0.516919
## STYPE2 1.372e+00 4.938e+00 0.278 0.781090
## STYPE3 4.516e-01 9.240e+00 0.049 0.961025
## STYPE4 2.763e-01 2.085e+01 0.013 0.989427
## STYPEE 1.063e-01 4.738e+00 0.022 0.982096
## STYPEH 4.048e+00 5.705e+00 0.709 0.478061
## STYPEM 5.841e+00 5.283e+00 1.106 0.268877
## VALID_NUM -7.991e-01 3.561e-01 -2.244 0.024862 *
## AA_NUM -3.045e-02 2.121e-02 -1.436 0.151152
## AA_SIGYes -3.391e+00 2.288e+00 -1.482 0.138345
## AI_NUM -6.781e-02 4.610e-02 -1.471 0.141362
## AI_SIGYes -8.093e-01 8.016e+00 -0.101 0.919580
## AS_NUM -4.403e-02 2.110e-02 -2.086 0.037018 *
## AS_SIGYes -1.582e+00 2.395e+00 -0.661 0.508908
## FI_NUM -5.377e-02 2.249e-02 -2.390 0.016861 *
## FI_SIGYes -3.005e+00 3.826e+00 -0.785 0.432197
## HI_NUM -4.091e-02 2.060e-02 -1.986 0.047085 *
## HI_SIGYes 2.237e+00 2.104e+00 1.063 0.287824
## PI_NUM -4.308e-02 5.160e-02 -0.835 0.403800
## PI_SIGYes -1.673e+01 9.627e+00 -1.738 0.082243 .
## WH_NUM -4.066e-02 2.077e-02 -1.958 0.050328 .
## WH_SIGYes 5.455e+00 1.823e+00 2.992 0.002781 **
## SD_NUM -4.120e-03 2.866e-03 -1.437 0.150669
## SD_SIGYes 1.189e+00 2.205e+00 0.539 0.589895
## EL_NUM 3.784e-03 3.014e-03 1.256 0.209326
## EL_SIGYes -4.284e+00 1.925e+00 -2.225 0.026102 *
## DI_NUM 7.052e-03 1.066e-02 0.662 0.508224
## DI_SIGYes 3.047e+00 1.991e+00 1.531 0.125913
## PCT_AA 4.742e-01 1.864e-01 2.544 0.010969 *
## PCT_AI 1.370e-01 2.284e-01 0.600 0.548628
## PCT_AS 2.192e+00 1.919e-01 11.422 < 2e-16 ***
## PCT_FI 1.622e+00 2.558e-01 6.338 2.48e-10 ***
## PCT_HI 1.148e+00 1.771e-01 6.483 9.61e-11 ***
## PCT_PI 5.823e-01 5.605e-01 1.039 0.298873
## PCT_WH 1.537e+00 1.770e-01 8.684 < 2e-16 ***
## MEALS -4.013e-01 4.415e-02 -9.088 < 2e-16 ***
## P_FDAY 9.904e-02 5.646e-02 1.754 0.079454 .
## P_GATE 1.141e+00 8.114e-02 14.062 < 2e-16 ***
## P_MIGED -4.586e-01 7.743e-02 -5.923 3.32e-09 ***
## P_EL -6.977e-01 6.889e-02 -10.127 < 2e-16 ***
## P_RFEP 8.360e-01 1.051e-01 7.956 2.08e-15 ***
## P_DI -1.358e+00 7.909e-02 -17.171 < 2e-16 ***
## SMOB -1.912e+00 4.606e-01 -4.151 3.35e-05 ***
## CBMOB 3.362e-01 4.649e-01 0.723 0.469513
## DMOB 2.913e-01 1.026e-01 2.839 0.004536 **
## PCT_RESP 2.379e-01 3.037e-02 7.835 5.45e-15 ***
## NOT_HSG -6.839e-01 1.498e-01 -4.565 5.08e-06 ***
## HSG -3.870e-01 1.069e-01 -3.621 0.000295 ***
## SOME_COL -4.914e-01 1.017e-01 -4.831 1.39e-06 ***
## COL_GRAD 1.297e-01 1.294e-01 1.002 0.316350
## GRAD_SCH 6.633e-01 1.780e-01 3.726 0.000196 ***
## AVG_ED 1.448e+01 6.414e+00 2.258 0.023997 *
## FULL 2.181e-01 8.914e-02 2.447 0.014438 *
## EMER 2.709e-02 1.222e-01 0.222 0.824590
## PEN_2 -2.330e+00 1.519e+00 -1.534 0.125145
## PEN_35 -1.954e+00 1.520e+00 -1.286 0.198534
## PEN_6 -2.668e+00 1.521e+00 -1.754 0.079396 .
## PEN_78 -2.391e+00 1.520e+00 -1.573 0.115757
## PEN_911 -2.663e+00 1.523e+00 -1.749 0.080374 .
## ENROLL -9.971e-02 3.069e-02 -3.249 0.001163 **
## PARENT_OPT 2.669e-01 9.873e-02 2.703 0.006887 **
## TESTED 1.058e-01 3.254e-02 3.253 0.001149 **
## VCST_E28 -3.215e-02 4.189e-01 -0.077 0.938814
## PCST_E28 NA NA NA NA
## VCST_E911 4.933e-01 1.563e-01 3.155 0.001613 **
## PCST_E911 NA NA NA NA
## CW_CSTE 8.870e+00 7.804e+00 1.137 0.255732
## VCST_M28 8.547e-01 2.769e-01 3.086 0.002035 **
## PCST_M28 NA NA NA NA
## VCST_M911 3.650e-01 3.518e-01 1.038 0.299519
## PCST_M911 NA NA NA NA
## CW_CSTM -5.585e+00 7.878e+00 -0.709 0.478347
## VCST_S28 1.336e-02 2.900e-02 0.461 0.645146
## PCST_S28 NA NA NA NA
## VCST_S911 -2.576e-03 9.824e-02 -0.026 0.979085
## PCST_S911 NA NA NA NA
## CW_CSTS -4.500e-01 7.616e+00 -0.059 0.952882
## VCST_H28 -3.701e-02 1.998e-02 -1.852 0.064057 .
## PCST_H28 NA NA NA NA
## VCST_H911 -1.312e-02 2.249e-02 -0.583 0.559677
## PCST_H911 NA NA NA NA
## CW_CSTH -3.998e-01 7.626e+00 -0.052 0.958187
## VNRT_R28 -4.921e+00 6.898e+00 -0.713 0.475597
## PNRT_R28 NA NA NA NA
## CW_NRTR 1.778e+00 1.795e+01 0.099 0.921081
## VNRT_L28 -1.075e+11 1.158e+11 -0.928 0.353298
## PNRT_L28 3.585e+12 3.862e+12 0.928 0.353298
## CW_NRTL 1.201e+02 4.214e+01 2.850 0.004389 **
## VNRT_S28 6.399e-01 7.996e-01 0.800 0.423522
## PNRT_S28 NA NA NA NA
## CW_NRTS -1.550e+02 4.179e+01 -3.710 0.000209 ***
## VNRT_M28 -2.361e-01 7.557e-01 -0.312 0.754724
## PNRT_M28 NA NA NA NA
## CW_NRTM 9.091e+00 1.605e+01 0.566 0.571121
## VCHS_E911 -6.596e-02 9.927e-02 -0.664 0.506404
## PCHS_E911 NA NA NA NA
## CW_CHSE 2.223e+00 7.674e+00 0.290 0.772121
## VCHS_M911 5.043e-02 9.471e-02 0.532 0.594406
## PCHS_M911 NA NA NA NA
## CW_CHSM 4.562e+00 7.642e+00 0.597 0.550600
## TOT_28 NA NA NA NA
## TOT_911 NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 45.11 on 6582 degrees of freedom
## Multiple R-squared: 0.8452, Adjusted R-squared: 0.8432
## F-statistic: 417.9 on 86 and 6582 DF, p-value: < 2.2e-16plot(lm)
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
## Warning in sqrt(crit * p * (1 - hh)/hh): NaNs produced
There’s a lot of NAs of Beta, which implies that it’s not estimatable.
We are going to perform variable selection using regsubsets-backward
regsubsets - model selection - backward - nvmax=10
Model Selection through stepwise backward selection from regsubsets
I let the maximum number of predictors be 10, since it’s going to be too complex to interpret or to build a model if we have more than 10 predictors
reg1<-regsubsets(API05B~., data=training.newbase, really.big=TRUE,nvmax=10, method = "backward")## Warning in leaps.setup(x, y, wt = wt, nbest = nbest, nvmax = nvmax,
## force.in = force.in, : 16 linear dependencies found## Reordering variables and trying again:reg.summary<-summary(reg1)
reg.summary## Subset selection object
## Call: regsubsets.formula(API05B ~ ., data = training.newbase, really.big = TRUE,
## nvmax = 10, method = "backward")
## 101 Variables (and intercept)
## Forced in Forced out
## STYPE2 FALSE FALSE
## STYPE3 FALSE FALSE
## STYPE4 FALSE FALSE
## STYPEE FALSE FALSE
## STYPEH FALSE FALSE
## STYPEM FALSE FALSE
## VALID_NUM FALSE FALSE
## AA_NUM FALSE FALSE
## AA_SIGYes FALSE FALSE
## AI_NUM FALSE FALSE
## AI_SIGYes FALSE FALSE
## AS_NUM FALSE FALSE
## AS_SIGYes FALSE FALSE
## FI_NUM FALSE FALSE
## FI_SIGYes FALSE FALSE
## HI_NUM FALSE FALSE
## HI_SIGYes FALSE FALSE
## PI_NUM FALSE FALSE
## PI_SIGYes FALSE FALSE
## WH_NUM FALSE FALSE
## WH_SIGYes FALSE FALSE
## SD_NUM FALSE FALSE
## SD_SIGYes FALSE FALSE
## EL_NUM FALSE FALSE
## EL_SIGYes FALSE FALSE
## DI_NUM FALSE FALSE
## DI_SIGYes FALSE FALSE
## PCT_AA FALSE FALSE
## PCT_AI FALSE FALSE
## PCT_AS FALSE FALSE
## PCT_FI FALSE FALSE
## PCT_HI FALSE FALSE
## PCT_PI FALSE FALSE
## PCT_WH FALSE FALSE
## MEALS FALSE FALSE
## P_FDAY FALSE FALSE
## P_GATE FALSE FALSE
## P_MIGED FALSE FALSE
## P_EL FALSE FALSE
## P_RFEP FALSE FALSE
## P_DI FALSE FALSE
## SMOB FALSE FALSE
## CBMOB FALSE FALSE
## DMOB FALSE FALSE
## PCT_RESP FALSE FALSE
## NOT_HSG FALSE FALSE
## HSG FALSE FALSE
## SOME_COL FALSE FALSE
## COL_GRAD FALSE FALSE
## GRAD_SCH FALSE FALSE
## AVG_ED FALSE FALSE
## FULL FALSE FALSE
## EMER FALSE FALSE
## PEN_2 FALSE FALSE
## PEN_35 FALSE FALSE
## PEN_6 FALSE FALSE
## PEN_78 FALSE FALSE
## PEN_911 FALSE FALSE
## ENROLL FALSE FALSE
## PARENT_OPT FALSE FALSE
## TESTED FALSE FALSE
## VCST_E28 FALSE FALSE
## VCST_E911 FALSE FALSE
## CW_CSTE FALSE FALSE
## VCST_M28 FALSE FALSE
## VCST_M911 FALSE FALSE
## CW_CSTM FALSE FALSE
## VCST_S28 FALSE FALSE
## VCST_S911 FALSE FALSE
## CW_CSTS FALSE FALSE
## VCST_H28 FALSE FALSE
## VCST_H911 FALSE FALSE
## CW_CSTH FALSE FALSE
## VNRT_R28 FALSE FALSE
## CW_NRTR FALSE FALSE
## VNRT_L28 FALSE FALSE
## CW_NRTL FALSE FALSE
## VNRT_S28 FALSE FALSE
## CW_NRTS FALSE FALSE
## VNRT_M28 FALSE FALSE
## CW_NRTM FALSE FALSE
## VCHS_E911 FALSE FALSE
## CW_CHSE FALSE FALSE
## VCHS_M911 FALSE FALSE
## CW_CHSM FALSE FALSE
## PCST_E28 FALSE FALSE
## PCST_E911 FALSE FALSE
## PCST_M28 FALSE FALSE
## PCST_M911 FALSE FALSE
## PCST_S28 FALSE FALSE
## PCST_S911 FALSE FALSE
## PCST_H28 FALSE FALSE
## PCST_H911 FALSE FALSE
## PNRT_R28 FALSE FALSE
## PNRT_L28 FALSE FALSE
## PNRT_S28 FALSE FALSE
## PNRT_M28 FALSE FALSE
## PCHS_E911 FALSE FALSE
## PCHS_M911 FALSE FALSE
## TOT_28 FALSE FALSE
## TOT_911 FALSE FALSE
## 1 subsets of each size up to 11
## Selection Algorithm: backward
## STYPE2 STYPE3 STYPE4 STYPEE STYPEH STYPEM VALID_NUM AA_NUM
## 1 ( 1 ) " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " " " " " " "
## 5 ( 1 ) " " " " " " " " " " " " " " " "
## 6 ( 1 ) " " " " " " " " " " " " " " " "
## 7 ( 1 ) " " " " " " " " " " " " " " " "
## 8 ( 1 ) " " " " " " " " " " " " " " " "
## 9 ( 1 ) " " " " " " " " " " " " " " " "
## 10 ( 1 ) " " " " " " " " " " " " " " " "
## 11 ( 1 ) " " " " " " " " " " " " " " " "
## AA_SIGYes AI_NUM AI_SIGYes AS_NUM AS_SIGYes FI_NUM FI_SIGYes
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " " " " "
## 5 ( 1 ) " " " " " " " " " " " " " "
## 6 ( 1 ) " " " " " " " " " " " " " "
## 7 ( 1 ) " " " " " " " " " " " " " "
## 8 ( 1 ) " " " " " " " " " " " " " "
## 9 ( 1 ) " " " " " " " " " " " " " "
## 10 ( 1 ) " " " " " " " " " " " " " "
## 11 ( 1 ) " " " " " " " " " " " " " "
## HI_NUM HI_SIGYes PI_NUM PI_SIGYes WH_NUM WH_SIGYes SD_NUM
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " " " " "
## 5 ( 1 ) " " " " " " " " " " " " " "
## 6 ( 1 ) " " " " " " " " " " " " " "
## 7 ( 1 ) " " " " " " " " " " " " " "
## 8 ( 1 ) " " " " " " " " " " " " " "
## 9 ( 1 ) " " " " " " " " " " " " " "
## 10 ( 1 ) " " " " " " " " " " " " " "
## 11 ( 1 ) " " " " " " " " " " " " " "
## SD_SIGYes EL_NUM EL_SIGYes DI_NUM DI_SIGYes PCT_AA PCT_AI PCT_AS
## 1 ( 1 ) " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " " " " " " "
## 5 ( 1 ) " " " " " " " " " " " " " " "*"
## 6 ( 1 ) " " " " " " " " " " " " " " "*"
## 7 ( 1 ) " " " " " " " " " " " " " " "*"
## 8 ( 1 ) " " " " " " " " " " " " " " "*"
## 9 ( 1 ) " " " " " " " " " " " " " " "*"
## 10 ( 1 ) " " " " " " " " " " " " " " "*"
## 11 ( 1 ) " " " " " " " " " " " " " " "*"
## PCT_FI PCT_HI PCT_PI PCT_WH MEALS P_FDAY P_GATE P_MIGED P_EL
## 1 ( 1 ) " " " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " "*" " " " " " " " "
## 3 ( 1 ) " " " " " " " " "*" " " " " " " " "
## 4 ( 1 ) " " " " " " " " "*" " " " " " " " "
## 5 ( 1 ) " " " " " " " " "*" " " " " " " " "
## 6 ( 1 ) " " " " " " "*" "*" " " " " " " " "
## 7 ( 1 ) " " " " " " "*" "*" " " " " " " " "
## 8 ( 1 ) " " " " " " "*" "*" " " " " " " " "
## 9 ( 1 ) " " " " " " "*" "*" " " "*" " " " "
## 10 ( 1 ) " " " " " " "*" "*" " " "*" " " " "
## 11 ( 1 ) " " " " " " "*" "*" " " "*" " " " "
## P_RFEP P_DI SMOB CBMOB DMOB PCT_RESP NOT_HSG HSG SOME_COL
## 1 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 2 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 3 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 4 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 5 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 6 ( 1 ) " " " " "*" " " " " " " " " " " " "
## 7 ( 1 ) " " "*" "*" " " " " " " " " " " " "
## 8 ( 1 ) " " "*" "*" " " " " " " " " " " " "
## 9 ( 1 ) " " "*" "*" " " " " " " " " " " " "
## 10 ( 1 ) " " "*" "*" " " " " " " " " " " " "
## 11 ( 1 ) " " "*" "*" " " " " " " "*" " " " "
## COL_GRAD GRAD_SCH AVG_ED FULL EMER PEN_2 PEN_35 PEN_6 PEN_78
## 1 ( 1 ) " " " " " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " " " " " "
## 4 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 5 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 6 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 7 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 8 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 9 ( 1 ) " " "*" " " " " " " " " " " " " " "
## 10 ( 1 ) " " "*" " " " " " " " " " " "*" " "
## 11 ( 1 ) " " "*" " " " " " " " " " " "*" " "
## PEN_911 ENROLL PARENT_OPT TESTED VCST_E28 PCST_E28 VCST_E911
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " " " " "
## 5 ( 1 ) " " " " " " " " " " " " " "
## 6 ( 1 ) " " " " " " " " " " " " " "
## 7 ( 1 ) " " " " " " " " " " " " " "
## 8 ( 1 ) " " " " " " " " " " " " " "
## 9 ( 1 ) " " " " " " " " " " " " " "
## 10 ( 1 ) " " " " " " " " " " " " " "
## 11 ( 1 ) " " " " " " " " " " " " " "
## PCST_E911 CW_CSTE VCST_M28 PCST_M28 VCST_M911 PCST_M911 CW_CSTM
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## VCST_S28 PCST_S28 VCST_S911 PCST_S911 CW_CSTS VCST_H28 PCST_H28
## 1 ( 1 ) " " " " " " " " " " " " " "
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## VCST_H911 PCST_H911 CW_CSTH VNRT_R28 PNRT_R28 CW_NRTR VNRT_L28
## 1 ( 1 ) " " " " " " " " " " " " " "
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## PNRT_L28 CW_NRTL VNRT_S28 PNRT_S28 CW_NRTS VNRT_M28 PNRT_M28
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
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## CW_NRTM VCHS_E911 PCHS_E911 CW_CHSE VCHS_M911 PCHS_M911 CW_CHSM
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
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## 8 ( 1 ) " " " " " " " " " " " " "*"
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## 11 ( 1 ) " " " " " " " " " " " " "*"
## TOT_28 TOT_911
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## 11 ( 1 ) " " " "I am going to see the changes of R^2, Adjusted R^2 (how well the model explained), Residuals Sum of Sqaured (Residuals), Cp (size of the bias), BIC
R Squared
plot(reg.summary$rsq, xlab="number of variables", ylab="Rsquared", type="l")
max(reg.summary$rsq)## [1] 0.8201761which.max(reg.summary$rsq)## [1] 11points(3, reg.summary$rsq[3], col="red",cex=1,pch=20)
points(4, reg.summary$rsq[4], col="red",cex=1,pch=20)
As the graph shows, we can notice a significant change of R^2 at 3 and 4 of number of variables
After that, we see the graph is slightly increasing
I assume that those points are the critical points.
Residuals Sum of Squared
plot(reg.summary$rss, xlab="number of variables", ylab="RSS", type="l")
min(reg.summary$rss)## [1] 15558564which.min(reg.summary$rss)## [1] 11points(3, reg.summary$rss[3], col="red",cex=1,pch=20)
points(4, reg.summary$rss[4], col="red",cex=1,pch=20)
Adjusted R Squared
plot(reg.summary$adjr2, xlab="number of variables", ylab="Adjusted R^2", type="l")
max(reg.summary$adjr2)## [1] 0.8198789which.max(reg.summary$adjr2)## [1] 11points(3,reg.summary$adjr2[3],col="red",cex=1,pch=20)
points(4,reg.summary$adjr2[4],col="red",cex=1,pch=20)
CP
From Penn State Univ Stat online Course..
“An underspecified model is a model in which important predictors are missing. And, an underspecified model yields biased regression coefficients and biased predictions of the response. Well, in short, Mallows’ CP statistic estimates the size of the bias that is introduced into the predicted responses by having an underspecified model.”
In short, CP implies the size of the bias
plot(reg.summary$cp, xlab="number of variables", ylab="Cp", type="l")
min(reg.summary$cp)## [1] 982.5362which.min(reg.summary$cp)## [1] 11points(3, reg.summary$cp[3],col="red",cex=1,pch=20)
points(4, reg.summary$cp[4],col="red",cex=1,pch=20)
BIC
plot(reg.summary$bic, xlab="number of variables", ylab="BIC", type="l")
min(reg.summary$bic)## [1] -11336.85which.min(reg.summary$bic)## [1] 11points(3, reg.summary$bic[3],col="red",cex=1,pch=20)
points(4, reg.summary$bic[4],col="red",cex=1,pch=20)
We would try to choose 4 number of predictors for our model from regsubsets.
a<-c("MEALS", "SMOB","GRAD_SCH","CW_CSTE")
reg.summary$rsq[4]## [1] 0.7507709reg.summary$adjr2[4]## [1] 0.7506213training.newbase1<-select(training.newbase,c("API05B",a))
rownames(training.newbase1)<-1:nrow(training.newbase1)Plots between dependent variable and independent variables
plot(training.newbase1)
GRAD_SCH with others shows some curves, it looks like log or squared term’s graph
training.newbase1 %>% keep(is.numeric) %>% gather() %>%
ggplot(aes(value)) +
facet_wrap(~key, scales="free")+
geom_histogram(bins=30)+
geom_density()
lm1<-lm(API05B~., data=training.newbase1)
summary(lm1)##
## Call:
## lm(formula = API05B ~ ., data = training.newbase1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -540.30 -27.88 3.36 31.79 342.30
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 621.06084 4.70077 132.12 <2e-16 ***
## MEALS -1.52129 0.03151 -48.29 <2e-16 ***
## SMOB -2.90255 0.05784 -50.18 <2e-16 ***
## GRAD_SCH 2.26144 0.07651 29.56 <2e-16 ***
## CW_CSTE 3.78087 0.08807 42.93 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 56.88 on 6664 degrees of freedom
## Multiple R-squared: 0.7508, Adjusted R-squared: 0.7506
## F-statistic: 5019 on 4 and 6664 DF, p-value: < 2.2e-16plot(lm1)
Adj R^2 75%, and there looks some patterns and some outliers
a<-predict(lm1,newdata=test.newbase)
b<-test.newbase$API05B
plot(a,b)
abline(0,1,col="red")
Predicted values vs test values of API05B
The graph looks like linear, and some outliers
predictions1 <- lm1 %>% predict(test.newbase)Model performance
data.frame(
MSE = mse(test.newbase$API05B, predictions1),
RMSE = RMSE(predictions1, test.newbase$API05B),
R2 = R2(predictions1, test.newbase$API05B)
)## MSE RMSE R2
## 1 3097.677 55.65678 0.75993591. Outliers
Finding outliers in our model
There are several ways to see outliers in our model
outlierTest(lm1,n.max=50,cutoff=0.05)## rstudent unadjusted p-value Bonferonni p
## 3762 -9.564767 1.5455e-21 1.0307e-17
## 1356 -6.105577 1.0819e-09 7.2152e-06
## 4812 -6.078444 1.2806e-09 8.5402e-06
## 2258 6.049796 1.5289e-09 1.0196e-05
## 3944 -5.831300 5.7572e-09 3.8395e-05
## 3364 -5.468623 4.7001e-08 3.1345e-04
## 743 -5.353180 8.9307e-08 5.9559e-04
## 2410 5.286745 1.2847e-07 8.5676e-04
## 2700 -5.187371 2.1958e-07 1.4644e-03
## 1452 -5.084014 3.7962e-07 2.5317e-03
## 1573 -4.945453 7.7827e-07 5.1903e-03
## 4680 -4.837192 1.3464e-06 8.9795e-03
## 4743 -4.756764 2.0086e-06 1.3396e-02
## 2314 -4.695724 2.7098e-06 1.8072e-02
## 488 -4.675050 2.9967e-06 1.9985e-02
## 2646 4.674809 3.0002e-06 2.0008e-02
## 5849 -4.635340 3.6314e-06 2.4218e-02
## 1118 4.612801 4.0470e-06 2.6990e-02
## 2054 4.512308 6.5227e-06 4.3500e-02
## 2143 -4.484455 7.4326e-06 4.9568e-02outlierTest shows which observations have greater than absolute value of 4 of rstudent.
qqPlot(lm1,main="QQ Plot")
## [1] 1356 3762qqPlot shows which observations have an impact on normality of residuals - which is outliers or bad leverage points
influencePlot(lm1,id.method="identify",main="influential plot",sub="circle size is proportial to cook's distance")## Warning in plot.window(...): "id.method" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "id.method" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "id.method" is
## not a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "id.method" is
## not a graphical parameter## Warning in box(...): "id.method" is not a graphical parameter## Warning in title(...): "id.method" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "id.method" is not a
## graphical parameter
## StudRes Hat CookD
## 1356 -6.105577 0.0041423445 0.030844268
## 2158 -3.624570 0.0117503290 0.031184299
## 2258 6.049796 0.0053462040 0.039135526
## 3762 -9.564767 0.0004855073 0.008768559
## 5582 -3.671075 0.0117526656 0.031994464Hat Values against Studentized Residuals
However,
I’m using Standardized Residuals to detect outliers
From Springers Text book “Linear Regression”, Chapter 3.2 - Regression Diagnostics: Tools for Checking the Validity of a Model, pg 60.
“In summary, an outlier is a point whose standardized residual falls outside the interval from -2 to 2. Recall that a bad leverage point is a leverage point which is also an outlier. Thus, a bad leverage point is a leverage point whose standardized residual falls outside the interval from -2 to 2.”
In very large datasets, we apply the rule to -4 to 4.
o1<-which(rstandard(lm1, infl = lm.influence(lm1, do.coef = FALSE),
sd=sqrt(deviance(lm1)/df.residual(lm1)),
type=c("sd.1","predictive"))>4)
o2<-which(rstandard(lm1, infl = lm.influence(lm1, do.coef = FALSE),
sd=sqrt(deviance(lm1)/df.residual(lm1)),
type=c("sd.1","predictive"))<(-4))
outliers <- c(o1,o2)
length(outliers)## [1] 36The data set after removing outliers
training.newbase2<-training.newbase1[-outliers,]Investigating linear models between response and each predictors - partial regression
API05B = B0 + B1*MEALS + E
lm.test1<-lm(API05B~MEALS, data=training.newbase2)
summary(lm.test1)##
## Call:
## lm(formula = API05B ~ MEALS, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -459.85 -32.83 18.05 58.69 260.33
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 821.21233 2.23874 366.82 <2e-16 ***
## MEALS -2.02565 0.03833 -52.84 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 94.34 on 6631 degrees of freedom
## Multiple R-squared: 0.2963, Adjusted R-squared: 0.2962
## F-statistic: 2792 on 1 and 6631 DF, p-value: < 2.2e-16plot(lm.test1)
Adj R^2 29.6%, plots looks ok except for normal Q-Q plot
API05B = B0 + B1*SMOB + E
lm.test2<-lm(API05B~SMOB, data=training.newbase2)
summary(lm.test2)##
## Call:
## lm(formula = API05B ~ SMOB, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -323.53 -59.63 -2.25 60.70 351.42
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 774.11173 1.33456 580.05 <2e-16 ***
## SMOB -4.95317 0.07364 -67.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 86.71 on 6631 degrees of freedom
## Multiple R-squared: 0.4056, Adjusted R-squared: 0.4055
## F-statistic: 4524 on 1 and 6631 DF, p-value: < 2.2e-16plot(lm.test2)
Adj R^2 40%, seems heteroscedasticity
API05B = B0 + B1*GRAD_SCH + E
lm.test3<-lm(API05B~GRAD_SCH, data=training.newbase2)
summary(lm.test3)##
## Call:
## lm(formula = API05B ~ GRAD_SCH, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -423.04 -38.67 10.97 56.64 288.02
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 660.58650 1.40222 471.10 <2e-16 ***
## GRAD_SCH 5.69449 0.08647 65.86 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 87.44 on 6631 degrees of freedom
## Multiple R-squared: 0.3954, Adjusted R-squared: 0.3953
## F-statistic: 4337 on 1 and 6631 DF, p-value: < 2.2e-16plot(lm.test3)
R^2 39.5%, plots shows extreme heteroscedasticity
API05B = B0 + B1*CW_CSTE + E
lm.test4<-lm(API05B~CW_CSTE, data=training.newbase2)
summary(lm.test4)##
## Call:
## lm(formula = API05B ~ CW_CSTE, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -411.25 -69.79 0.66 70.67 326.49
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 456.9592 6.2326 73.32 <2e-16 ***
## CW_CSTE 5.4549 0.1268 43.03 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 99.43 on 6631 degrees of freedom
## Multiple R-squared: 0.2183, Adjusted R-squared: 0.2182
## F-statistic: 1852 on 1 and 6631 DF, p-value: < 2.2e-16plot(lm.test4)
R^2 21.8%, plots shows some patterns showing heteroscedasticity
Model after removing outliers
lm2<-lm(API05B~., data=training.newbase2)
summary(lm2)##
## Call:
## lm(formula = API05B ~ ., data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -225.373 -28.309 2.242 30.659 230.408
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 623.13764 4.40548 141.45 <2e-16 ***
## MEALS -1.56062 0.02955 -52.81 <2e-16 ***
## SMOB -2.95763 0.05448 -54.29 <2e-16 ***
## GRAD_SCH 2.21507 0.07178 30.86 <2e-16 ***
## CW_CSTE 3.81710 0.08252 46.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 53.11 on 6628 degrees of freedom
## Multiple R-squared: 0.777, Adjusted R-squared: 0.7769
## F-statistic: 5775 on 4 and 6628 DF, p-value: < 2.2e-16plot(lm2)
Adj R^2 77%, plots looks ok except for showing some pattern of the heteroscedasticity.
R^2 getting higher, plots for model such as ‘fitted values vs residuals’, ‘Normal Q-Q’, or ‘sqrt of Standardized residuals vs fitted values’ becomes much better after removing outliers
It’s still showing some patterns, I will perform diagnosis of normality of residuals, multicollinearity, and heteroscedasticity from now.
a1<-predict(lm2,newdata = test.newbase)
predictions2 <- lm2 %>% predict(test.newbase)Model performance
data.frame(
MSE = mse(test.newbase$API05B, predictions2),
RMSE = RMSE(predictions2, test.newbase$API05B),
R2 = R2(predictions2, test.newbase$API05B)
)## MSE RMSE R2
## 1 3097.117 55.65174 0.75987072. Normality (of residuals)
We don’t have to worry about the assumption of normality of residuals (the error between the dependent variable and the independent variables) because when the sample size is sufficiently large, the Central Limit Theorem ensures that the distribution of residiual will be approximately normality.
qplot(residuals(lm2),
geom="histogram",
binwidth= 10,
main="Histogram of Residuals of our model",
xlab="Residuals of our model",
ylab="Frequency")
mean(residuals(lm2))## [1] -8.158894e-16Residuals on our model are approximately normal distributed with zero mean.
3. Multicolliniearity
From STHDA website..
definition
There is an extreme situation, called multicollinearity, where collinearity exists between three or more variables even if no pair of variables has a particularly high correlation. This means that there is redundancy between predictor variables."
cor<-cor(training.newbase2)
corrplot(cor,
method=c("circle"),
title="Correlation between variables",
type=c("full"))
vif(lm2)## MEALS SMOB GRAD_SCH CW_CSTE
## 1.874570 1.458532 1.867840 1.485354More than 4 VIF score must be removed, but we dont have any.
4. Heteroscedasciticity (non-constant residuals)
plot(lm2)
As we see the first plot of the model (fitted value vs residuals),
the residuals plots show decreasing variance towards the right end.
It seems like there’s heteroscedasticity exists, so I’m going to perform transformation.
Box-cox
bc<-boxcox(lm2)
Finding maximum log-likelihoods for the parameter of the Box-Cox power trasnformation, which is lambda
lambda<-bc$x[which.max(bc$y)]
lambda## [1] 2Boxcox power + log transformation
lm3<-lm(API05B^(lambda)~MEALS+log(SMOB+1)+log(GRAD_SCH+1)+CW_CSTE, data=training.newbase2)
summary(lm3)##
## Call:
## lm(formula = API05B^(lambda) ~ MEALS + log(SMOB + 1) + log(GRAD_SCH +
## 1) + CW_CSTE, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -327955 -44794 605 46969 327029
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398321.29 7812.70 50.98 <2e-16 ***
## MEALS -2127.65 44.86 -47.42 <2e-16 ***
## log(SMOB + 1) -61332.05 1362.81 -45.00 <2e-16 ***
## log(GRAD_SCH + 1) 35881.42 1350.71 26.57 <2e-16 ***
## CW_CSTE 6170.79 109.67 56.27 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 75390 on 6628 degrees of freedom
## Multiple R-squared: 0.7682, Adjusted R-squared: 0.7681
## F-statistic: 5491 on 4 and 6628 DF, p-value: < 2.2e-16plot(lm3)
Adj R^2 76.8%.
The residuals plots look improved, which is that it looks like now it’s homoscedasticity (constant residuals, no patterns)
a2<-sqrt(predict(lm3,newdata = test.newbase))Model Progress by showing graphs of “predicted vs test values”
plot(a,b,
main="First model prediction",
xlab="predicted values",
ylab="values in test dataset")
abline(a=0,b=1,col="red")
plot(a1,b,
main="Removed outliers",
xlab="predicted values",
ylab="values in test dataset")
abline(a=0,b=1,col="red")
plot(a2,b,
main="After transformation - Final model",
xlab="predicted values",
ylab="values in test dataset")
abline(a=0,b=1,col="red")
The graphs with final model seems to be linear, and there is no patters.
predictions3 <- lm3 %>% predict(test.newbase)Model performance
data.frame(
MSE = mse(test.newbase$API05B, predictions3),
RMSE = RMSE(predictions3, test.newbase$API05B),
R2 = R2(predictions3, test.newbase$API05B)
)## MSE RMSE R2
## 1 298721055021 546553.8 0.7477782Model performance changing process
c.m<-data.frame(MSE=
c(mse(test.newbase$API05B, predictions1),
mse(test.newbase$API05B, predictions2),
mse(test.newbase$API05B, predictions3)
),
RMSE=
c(RMSE(predictions1,test.newbase$API05B),
RMSE(predictions2,test.newbase$API05B),
RMSE(predictions3,test.newbase$API05B)
),
R2=
c(R2(predictions1,test.newbase$API05B),
R2(predictions2,test.newbase$API05B),
R2(predictions3,test.newbase$API05B)
))
rownames(c.m)<-c("First model","Removed Outliers","After transformation-final model")
c.m## MSE RMSE R2
## First model 3.097677e+03 55.65678 0.7599359
## Removed Outliers 3.097117e+03 55.65174 0.7598707
## After transformation-final model 2.987211e+11 546553.79884 0.7477782summary(lm3)##
## Call:
## lm(formula = API05B^(lambda) ~ MEALS + log(SMOB + 1) + log(GRAD_SCH +
## 1) + CW_CSTE, data = training.newbase2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -327955 -44794 605 46969 327029
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 398321.29 7812.70 50.98 <2e-16 ***
## MEALS -2127.65 44.86 -47.42 <2e-16 ***
## log(SMOB + 1) -61332.05 1362.81 -45.00 <2e-16 ***
## log(GRAD_SCH + 1) 35881.42 1350.71 26.57 <2e-16 ***
## CW_CSTE 6170.79 109.67 56.27 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 75390 on 6628 degrees of freedom
## Multiple R-squared: 0.7682, Adjusted R-squared: 0.7681
## F-statistic: 5491 on 4 and 6628 DF, p-value: < 2.2e-16Conclusion
I conclude that we are able to predict the API score for the future with the 3 of predictors, which are described at below, as 76.8% variance explained with all transformed predictors have less than 0.05 p-values, which is that they are all significant predictors on our response variable.
As interpretation of the Betas (each coefficients),
For example,
- For a 1 units increase in MEALS,
\(\mid(Beta1)\mid\) = 2127.65 decrease in \((API05B)^2\) score
- For a 1% increase in SMOB,
\(\mid Beta2\mid\) \(\times\) \({\log(1.01+1))}\) = 61332.05 \(\times\) \({\log(1.01+1))}\) = 42818.66
Therefore,
\(\mid Beta2\mid\) \(\times\) \({\log(1.01+1))}\) = 61332.05 \(\times\) \({\log(1.01+1))}\) = 42818.66 decrease in squared API05B score
In other words, 1 percent change in SMOB is associated with
\(\mid Beta2\mid\) \(\times\) \({\log(1.01+1))}\) change in \(API05B^(lambda)\) = \((API05B)^2\), where lambda is 2
- For a 10% increase in GRAD_SCH,
Beta3 \(\times\) \({\log (1.1+1)}\) = 35881.42 \(\times\) \({\log (1.1+1)}\) = 26621.77
API score is transformed with lambda, which is power to the 2, from box-cox transformation,
Therefore,
Beta3 \(\times\) \({\log(1.1+1))}\) = 35881.42 \(\times\) \({\log (1.1+1)}\) = 26621.77 increase in squared API05B score
In other words, 1 percent change in GRAD_SCH is associated with
Beta3 \(\times\) \({\log(1.1+1))}\) change in \(API05B^(lambda)\) = \((API05B)^2\), where lambda is 2
- For a 10 units increase in CW_CSTE,
Beta4 \(\times\) 10 = 6170.79 \(\times\) 10 = 61707.9 increase in sqaured API05B score